This assumption either means that you already assume that neither one of them is really a revelation from God, since you called it "tradition". Which would require an evidence on your behalf to prove that neither christianity nor Islam was really a revelation from God to true prophets.
That is not correct. Calling Judaism, Christianity and Islam traditions in no way implies anything about the truth of their revelations. All that it implies is that they have been maintained through various generations of keepers of some sort. Which is true of all three religions, at least according to themselves.
Or you actually have no idea what Islam or the Quran states. Where exactly does it say in the Quran that Islam came out of the "christian tradition" whatever that means?
Sura 19, for one particularly visible example.
Or better, yet:
Qur'an 3:55. And when Allah said: O Isa, [Jesus] I am going to terminate the period of your stay (on earth) and cause you to ascend unto Me and purify you of those who disbelieve and make those who follow you above those who disbelieve to the day of resurrection; then to Me shall be your return, so l will decide between you concerning that in which you differed.
And
Qur'an 4:155159. Then because of their breaking of their covenant, and their disbelieving in the revelations of Allah, and their slaying of the prophets wrongfully, and their saying: Our hearts are hardened Nay, but Allah set a seal upon them for their disbelief, so that they believe not save a few
And because of their disbelief and of their speaking against Mary a tremendous calumny;
And because of their saying: We slew the Messiah, Jesus son of Mary, Allah's messenger they slew him not nor crucified him, but it appeared so unto them; and lo! those who disagree concerning it are in doubt thereof; they have no knowledge thereof save pursuit of a conjecture; they slew him not for certain.
But Allah took him up unto Himself. Allah was ever Mighty, Wise.
There is not one of the People of the Scripture but will believe in him before his death, and on the Day of Resurrection he will be a witness against them.
Or to make it easier, please explain what do you mean by "came out of", and what do you mean by "christian tradition"?
I'm a bit surprised by your questions, myself. Christianity is refered so strongly and so emphatically in the Quran that I'm not even sure I understand what you are asking.